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arXiv · 2509.18940

Extending total colorings in planar graphs

Owen Henderschedt·Jessica McDonald
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Abstract

We initiate the study of total-coloring extensions, and focus our attention on planar graphs, asking: ``When can a total-$k$-coloring of some subgraph $H$ of a planar graph $G$ be extended to a total-$k$-coloring of $G$?'' We prove that if $H$ is a matching, then any total-$(\Delta+3)$-coloring of $H$ in $G$ extends to $G$ provided $\Delta\geq 28$; this number of colors is best-possible without introducing a distance condition on $H$. We also prove that if $H$ is a set of distance-3 cliques then any total-$(\Delta+1)$-coloring of $H$ extends to $G$ provided $\Delta\geq 27$; this distance condition cannot be lowered.

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Owen Henderschedt, Jessica McDonald. 2025-09-23. Extending total colorings in planar graphs. https://arxiv.org/abs/2509.18940

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